Addendum: Some Limit Theorems of Renewal Theory

1984 ◽  
Vol 28 (4) ◽  
pp. 856-857
Author(s):  
A. V. Nagaev
Keyword(s):  
Limit Theorems ◽  
Renewal Theory

Limit theorems for first-passage times in linear and non-linear renewal theory

1984 ◽  
Vol 16 (04) ◽  
pp. 766-803 ◽  
Author(s):  
S. P. Lalley
Keyword(s):  
Limit Theorems ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Renewal Theory ◽  
Local Limit ◽  
Linear Boundary ◽  
First Passage ◽  
Non Linear ◽  
First Time ◽  
Passage Times

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.

Limit theorems for sums of a sequence of random variables defined on a Markov chain

1977 ◽  
Vol 14 (03) ◽  
pp. 614-620
Author(s):  
David B. Wolfson
Keyword(s):  
Limit Theorems ◽  
Transition Probabilities ◽  
Sufficient Conditions ◽  
Renewal Theory ◽  
Strong Mixing ◽  
Mixing Condition ◽  
The Matrix ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Markov Renewal Theory

Let {(Jn, Xn),n≧ 0} be the standardJ–Xprocess of Markov renewal theory. Suppose {Jn,n≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that ifconverges in distribution, wherean, bn>0 (bn→∞) are real constants, then the limit lawFmust be stable. SupposeQ(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn),n≧ 0}. Then then-fold convolution,Q∗n(bnx + anbn), converges in distribution toF(x)Π if and only ifconverges in distribution toF. Π is the matrix of stationary transition probabilities of {Jn,n≧ 0}. Sufficient conditions on theHi's are given for the convergence of the sequence of semi-Markov matrices toF(x)Π, whereFis stable.

Limit theorems for first-passage times in linear and non-linear renewal theory

1984 ◽  
Vol 16 (4) ◽  
pp. 766-803 ◽  
Author(s):  
S. P. Lalley
Keyword(s):  
Limit Theorems ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Renewal Theory ◽  
Local Limit ◽  
Linear Boundary ◽  
First Passage ◽  
Non Linear ◽  
First Time ◽  
Passage Times

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.

scholarly journals Local limit theorems and renewal theory with no moments

2016 ◽  
Vol 21 (0) ◽  
Author(s):  
Kenneth S. Alexander ◽  
Quentin Berger
Keyword(s):  
Limit Theorems ◽  
Renewal Theory ◽  
Local Limit ◽  
Local Limit Theorems

Renewal theory in a random environment

1994 ◽  
Vol 116 (1) ◽  
pp. 179-190
Author(s):  
Laurence A. Baxter ◽  
Linxiong Li
Keyword(s):  
Stochastic Process ◽  
Random Environment ◽  
Limit Theorems ◽  
Random Variables ◽  
Renewal Theory ◽  
Renewal Processes ◽  
Conditional Distributions ◽  
The Future ◽  
Standard Limit

AbstractA random environment is modelled by an arbitrary stochastic process, the future of which is described by a σ-algebra. Renewal processes and alternating renewal processes are defined in this environment by considering the conditional distributions of random variables generated by the processes with respect to the σ-algebra. Generalizations of several of the standard limit theorems of renewal theory are derived.

Periodicity in Markov renewal theory

1974 ◽  
Vol 6 (1) ◽  
pp. 61-78 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Renewal Process ◽  
Limit Theorems ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Regenerative Processes ◽  
Markov Kernel ◽  
Transient States ◽  
Initial States ◽  
Markov Renewal Theory ◽  
Direct Inspection

In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems will be given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes.

scholarly journals Renewal theory for random walks on surface groups

2016 ◽  
Vol 38 (1) ◽  
pp. 155-179 ◽  
Author(s):  
PETER HAÏSSINSKY ◽  
PIERRE MATHIEU ◽  
SEBASTIAN MÜLLER
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Asymptotic Variance ◽  
Renewal Theory ◽  
Surface Groups ◽  
Exponential Moment ◽  
Moment Conditions

We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enter a particular type of cone and never leave it again. As a consequence, the trajectory of the random walk can be expressed as analigned unionof independent and identically distributed trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.

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